cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 11 results. Next

A244657 Number T(n,k) of n-node unlabeled rooted trees with thinning limbs and root outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 4, 3, 1, 1, 0, 1, 9, 6, 3, 1, 1, 0, 1, 13, 13, 6, 3, 1, 1, 0, 1, 26, 25, 15, 6, 3, 1, 1, 0, 1, 42, 55, 29, 15, 6, 3, 1, 1, 0, 1, 81, 107, 68, 31, 15, 6, 3, 1, 1, 0, 1, 138, 224, 140, 72, 31, 15, 6, 3, 1, 1
Offset: 1

Views

Author

Alois P. Heinz, Jul 04 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Examples

			The A124343(5) = 6 5-node rooted trees with thinning limbs sorted by root outdegree are:
:  o  :   o       o     o   :   o   :    o    :
:  |  :  / \     / \   / \  :  /|\  :  /( )\  :
:  o  : o   o   o   o o   o : o o o : o o o o :
:  |  : |      / \    |   | : |     :         :
:  o  : o     o   o   o   o : o     :         :
:  |  : |                   :       :         :
:  o  : o                   :       :         :
:  |  :                     :       :         :
:  o  :                     :       :         :
:     :                     :       :         :
: -1- : ---------2--------- : --3-- : ---4--- :
Thus row 5 = [0, 1, 3, 1, 1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  1,   1;
  0, 1,  3,   1,  1;
  0, 1,  4,   3,  1,  1;
  0, 1,  9,   6,  3,  1,  1;
  0, 1, 13,  13,  6,  3,  1, 1;
  0, 1, 26,  25, 15,  6,  3, 1, 1;
  0, 1, 42,  55, 29, 15,  6, 3, 1, 1;
  0, 1, 81, 107, 68, 31, 15, 6, 3, 1, 1;

Links

Crossrefs

Columns k=0-10 give: A000007(n-1), A000012 (for n>1), A244703, A244704, A244705, A244706, A244707, A244708, A244709, A244710, A244711.
T(2n,n) gives A244712.
Row sums give A124343.
Cf. A245120, A245151.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0,
      `if`(v=0, 1, 0), `if`(i<1 or v<1 or n b(n-1$2, k$2):
seq(seq(T(n, k), k=0..n-1), n=1..14);
  • Mathematica
    b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i<1 || v<1 || nJean-François Alcover, Feb 03 2015, after Alois P. Heinz *)

A124346 Number of rooted identity trees on n nodes with thinning limbs.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 11, 17, 32, 56, 102, 184, 340, 624, 1161, 2156, 4036, 7562, 14234, 26828, 50747, 96125, 182545, 347187, 661618, 1262583, 2413275, 4618571, 8850905, 16981142, 32616900, 62713951, 120703497, 232527392, 448344798, 865182999, 1670884073
Offset: 1

Views

Author

Christian G. Bower, Oct 30 2006, suggested by Franklin T. Adams-Watters

Keywords

Comments

A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Examples

			The a(7) = 6 trees are ((((((o)))))), (o((((o))))), (o(o((o)))), ((o)(((o)))), ((o)(o(o))), (o(o)((o))). - _Gus Wiseman_, Jan 25 2018

Links

Crossrefs

Cf. A000081, A001678, A003238, A004111, A124343-A124348, A276625, A290689, A298303-A298305, A298422.
Row sums of A245120.

Programs

  • Mathematica
    idthinQ[t_]:=And@@Cases[t,b_List:>UnsameQ@@b&&Length[b]>=Max@@Length/@b,{0,Infinity}];
itrut[n_]:=itrut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[itrut/@c]]]/@IntegerPartitions[n-1],idthinQ]];
Table[Length[itrut[n]],{n,25}] (* Gus Wiseman, Jan 25 2018 *)

A245121 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 2.

Original entry on oeis.org

1, 1, 3, 4, 8, 12, 22, 36, 63, 107, 188, 327, 578, 1020, 1820, 3248, 5839, 10511, 19022, 34484, 62755, 114421, 209234, 383327, 703901, 1294822, 2386376, 4405083, 8144701, 15080416, 27961728, 51912054, 96496481, 179577543, 334558479, 623936240, 1164765120
Offset: 4

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Examples

			a(7) = 4:
:   o     o       o      o   :
:  / \   / \     / \    / \  :
: o   o o   o   o   o  o   o :
: |     |   |  / \    ( )  | :
: o     o   o o   o   o o  o :
: |     |     |       |      :
: o     o     o       o      :
: |     |     |              :
: o     o     o              :
: |                          :
: o                          :

Links

Crossrefs

Column k=2 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 2$2):
seq(a(n), n=4..45);
  • Mathematica
    b[n_, i_, h_, v_] := b[n, i, h, v] = If[n == 0, If[v == 0, 1, 0], If[i < 1 || v < 1 || n < v, 0, Sum[Binomial[A[i, Min[i - 1, h]], j] b[n - i*j, i - 1, h, v - j], {j, 0, Min[n/i, v]}]]];
A[n_, k_] := A[n, k] = If[n<2, n, Sum[b[n-1, n-1, j, j], {j, 1, Min[k, n-1] } ] ];
a[n_] := b[n-1, n-1, 2, 2];
a /@ Range[4, 45] (* Jean-François Alcover, Dec 28 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 1.938950593419038561279875... and c = 0.929315638487153276953929... . - Vaclav Kotesovec, Jul 13 2014

A245122 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 3.

Original entry on oeis.org

1, 2, 4, 9, 17, 35, 67, 131, 249, 484, 922, 1775, 3393, 6513, 12461, 23910, 45806, 87903, 168619, 323796, 621816, 1195123, 2297617, 4420093, 8506487, 16380013, 31554394, 60817066, 117266799, 226217218, 436572777, 842895506, 1628036630, 3145780899, 6080759314
Offset: 7

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Comments

In a rooted tree with thinning limbs the outdegree of a parent node is larger than or equal to the outdegree of any of its child nodes.

Examples

			a(8) = 2:
:    o        o     :
:   /|\     / | \   :
:  o o o   o  o  o  :
:  | |    ( ) |     :
:  o o    o o o     :
:  |      |         :
:  o      o         :
:  |                :
:  o                :

Links

Crossrefs

Column k=3 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 3$2):
seq(a(n), n=7..45);

A245123 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 4.

Original entry on oeis.org

2, 3, 9, 20, 46, 94, 202, 412, 850, 1719, 3483, 6987, 14026, 27990, 55830, 111022, 220589, 437451, 866898, 1715821, 3393973, 6708016, 13251455, 26163174, 51635765, 101868226, 200908954, 396129137, 780868821, 1538971204, 3032575428, 5974874666, 11770477038
Offset: 11

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Examples

			a(11) = 2:
:     o          o      :
:   /( )\     / ( ) \   :
:  o o o o   o  o o  o  :
:  | | |    ( ) | |     :
:  o o o    o o o o     :
:  | |      |   |       :
:  o o      o   o       :
:  |                    :
:  o                    :

Links

Crossrefs

Column k=4 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 4$2):
seq(a(n), n=11..50);

A245124 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 5.

Original entry on oeis.org

1, 4, 11, 28, 70, 160, 366, 804, 1748, 3734, 7918, 16597, 34601, 71628, 147631, 302857, 619231, 1261849, 2564795, 5200248, 10522565, 21252174, 42854194, 86286963, 173517189, 348523105, 699311092, 1401837776, 2807733181, 5619221464, 11238041122, 22460777472
Offset: 15

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Examples

			a(15) = 1:
:        o        :
:   / (  |  ) \   :
:  o  o  o  o  o  :
:  | ( ) |  |     :
:  o o o o  o     :
:  | |   |        :
:  o o   o        :
:  |              :
:  o              :

Links

Crossrefs

Column k=5 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 5$2):
seq(a(n), n=15..50);

A245125 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 6.

Original entry on oeis.org

2, 6, 23, 60, 162, 397, 960, 2223, 5085, 11355, 25088, 54654, 118051, 252601, 536973, 1133925, 2382281, 4980512, 10370545, 21512821, 44483291, 91708748, 188580249, 386854596, 791909788, 1617922147, 3299701619, 6718766927, 13660421145, 27736326713, 56246087592
Offset: 20

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Links

Crossrefs

Column k=6 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 6$2):
seq(a(n), n=20..55);

A245126 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 7.

Original entry on oeis.org

1, 8, 26, 86, 247, 669, 1709, 4251, 10214, 24066, 55551, 126369, 283505, 629261, 1382778, 3013846, 6519955, 14015077, 29952488, 63690016, 134807361, 284170813, 596800591, 1249172169, 2606663357, 5424220543, 11258470062, 23313312932, 48171597034, 99337649116
Offset: 25

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Links

Crossrefs

Column k=7 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 7$2):
seq(a(n), n=25..60);

A245127 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 8.

Original entry on oeis.org

4, 18, 75, 241, 732, 2048, 5507, 14149, 35406, 86251, 206060, 483503, 1118366, 2553371, 5766634, 12896468, 28598143, 62934478, 137565845, 298871089, 645779488, 1388442085, 2971788670, 6334659311, 13452368784, 28469327221, 60059197787, 126331495950, 265014539903
Offset: 31

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Links

Crossrefs

Column k=8 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 8$2):
seq(a(n), n=31..65);

A245128 Number of n-node rooted identity trees with thinning limbs and root outdegree (branching factor) 9.

Original entry on oeis.org

6, 32, 140, 490, 1582, 4679, 13207, 35579, 92848, 235364, 583368, 1417164, 3386221, 7972754, 18536344, 42613503, 97001737, 218855237, 489889224, 1088756521, 2404139499, 5277595013, 11523611147, 25038756870, 54160808489, 116670213947, 250366691420, 535375247787
Offset: 37

Views

Author

Alois P. Heinz, Jul 12 2014

Keywords

Links

Crossrefs

Column k=9 of A245120.

Programs

  • Maple
    b:= proc(n, i, h, v) option remember; `if`(n=0, `if`(v=0, 1, 0),
      `if`(i<1 or v<1 or n b(n-1$2, 9$2):
seq(a(n), n=37..70);
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.